A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

Abstract

The investigation and quantification of cardiac motion is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider a new method to track cardiac motion from magnetic resonance (MR) tagged images. Tracking is achieved by following the spatial maxima in scale-space of the MR images over time. Reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev norm expressed in covariant derivatives. These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. Furthermore, we propose a multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in one nonsingular analytic kernel operator in order to decompose the optic flow vector field in a divergence-free and a rotation-free part. Finally, we combine both the multi-scale Helmholtz decomposition and our vector field reconstruction (based on covariant derivatives) in a single algorithm and show the practical benefit of this approach by an experiment on real cardiac images.

Evolution equations on Gabor transforms and their applications

“…For example, if a = 0 and D would be constant, then by Eq. (20) and Schur's lemma one has D = diag{D 11 , . .…”

“…Some of the optic flow methods, such as [55,56,57], do allow direct computation on the original MRI-tagging images as well, but they are relatively expensive, complicated and technical, e.g. [20].…”

“…Our approach by Gabor transforms is inspired by [19] and it is relatively simple compared to existing approaches on cardiac deformation (or strain/velocity) estimations, cf. [20,21,22,23].…”

Fluid Flow Estimation with Multiscale Ensemble Filters Based on Motion Measurements Under Location Uncertainty

“…The reader can refer to [16] for a recent review on estimators dedicated to fluid flows and more generally to [1,2,12] for presentation and comparative performance evaluations of some state-of-the-art techniques in computer vision. Interested readers may also refer to [3,7,24], which provide recent trends for data based adaptive deformation estimation schemes.…”

“…Several motion estimators dedicated to the measurement of specific phenomenon such as fluid flows have been proposed in the literature (see [16] for a detailed overview). These estimators differ mainly on the smoothness function they are handling: first order penalization [29], second order div-curl regularization [6,33], data-dependent [3,7,24] or power law auto-similarity principles [14,15]. These methods provide accurate instantaneous displacements as they generally implement additional constraints imposed by the physics, as in [6,11] and most of them are embedded into a multiscale formalism that enables capturing efficiently the large scales deformations [16].…”